Systems that kept the measure
What the removed roof buys
The Japanese convention of drawing an interior with its roof lifted off is usually explained as a way of seeing inside. What it actually buys is uniformity — every room reports the same share of its floor, to the last sample, where the eye that frames the same building reports three different numbers.
Measuring a room off the page
A perspective picture of a floor has to be rectified before anything on it can be measured, and the rectification is a fit that amplifies the marking error. An oblique picture of the same floor is already rectified — the page is the plan, at one scale, and a ruler on the paper is a ruler on the ground.
A picture with no size–distance signal
In a system with no diminution the drawn size of an object falls at exactly zero pixels per metre, so nothing in the picture says how far away anything is. Depth has to be carried by something else, and what carries it is height on the page — linearly, and without a horizon.
A carpet and the people on it
A Persian miniature draws the ground as though from above and the figures standing on it as though from in front. The two views want optical axes exactly ninety degrees apart, and the plan view does not shorten a standing figure — it replaces its height with its distance from the point under the eye.
Assembled from several views
An Egyptian relief takes each part of a figure from the direction that identifies it — head in profile, eye and shoulders frontal, a pond in plan. What that buys is exactly measurable: any single viewing direction keeps at most √k of k perpendicular aspects, so the best compromise view retains 58% of each.
A picture with two eyes in it
Several traditions draw the floor from one place and the people on it from another. No single camera produces both, as an earlier essay showed. What such a picture actually is has a measurement attached: give the rays their world points and ask for the one place they all pass through, and at a stride of separation the best answer misses them by six tenths of a metre.
The second eye is a shear
A picture drawn from two eyes is a picture drawn from one, of a different room. The map that puts the second eye away holds the picture plane still point by point and carries one centre onto the other, and the member of its family that matters turns out to be affine — a shear along the line joining the eyes, growing with depth, which is the same operation an oblique drawing performs.
Counting the eyes needs the room
How many eyes made a picture is not a question the picture can be asked. Told what the room really measures, the rays refuse to meet and a second eye has been caught; told instead that the room is the one the picture is consistent with, the same rays meet exactly, at the first eye. The refusal is real and it belongs to the room.
Two grounds, and what the second one costs
The miniature convention wants its floor drawn from overhead and its figures drawn from in front, and the two optical axes it asks for are exactly ninety degrees apart. Read as a picture with two centres rather than as a picture with none, the arrangement stops being a contradiction and becomes a quantity: the rays of the composite miss their own best point by more than a metre, and the absorbed reading is a floor that leans.
A parallel floor under a perspective room
Draw the floor without diminution and the people on it with it, and the picture has a centre at infinity glued to a centre in the room. The same map absorbs it — but there is no shear this time, and there cannot be: bringing a point in from infinity is not something an affine map does, so the room the picture is equally a picture of has its midpoints moved as well as its angles.
The same person, twice on one panel
A panel showing one figure at four moments is geometrically the least strange thing in this field — one camera, one floor, and every pair of copies meeting the horizon to 8.5 × 10⁻¹⁴ px. What the picture withholds is the order, and four copies admit twenty-four readings, and a reading convention supplies 4.6 bits from outside the marks. Enlarge one figure by six per cent and the horizon test that passed the panel catches it at 36 px.
Size that means rank
In a great many pictures the drawn height records importance rather than distance. That is a decision rather than a mistake, and it can be caught with a straightedge by carrying one figure's height across the room by the taught construction and see where it lands. A tenth of rank in the picture already misses the drawn head by 21 px, the miss is exactly linear in how much rank is there, and the whole test needs two references and no arithmetic.
A picture in bands
A register picture stacks its scene in horizontal bands, each with its own ground line and every figure drawn at one height. The feet line and the heads line of any pair are then parallel to the arithmetic floor — 0° against 15.4° in a photograph of the same figures — so the picture has no horizon anywhere in it, and what a reader recovers is an ordering with no metre attached.
The pond with its trees laid flat
An Egyptian garden pond is drawn in plan with its trees rotated outward about the bank they stand on. A rotation is an isometry, so every length in that drawing is exactly the length it is in the garden — zero error, not a small one. What is spent is the angle between any two faces, which reads 180° across every hinge and is 90° in the garden, and the four walls admit sixteen assemblies, so a reader supplies four bits to fold it back up.
What survives being copied
A workshop copying a drawing from a drawing is a random walk — the spread across lineages grows as the square root of the generation, with a fitted exponent of 0.5001. A workshop copying the method is not, and its exponent is 0.012, which is no growth at all, and after forty generations two lineages started from different originals end up 5 × 10⁻¹⁵ apart. Copying the marks loses the picture; copying the recipe loses the original and keeps the recipe.
The workshop that throws drawings away
Adding a rule that discards a drawing which looks wrong turns the mark copyist's spread from a growing random walk into a stationary process — the fitted exponent falls from 0.542 to 0.022, indistinguishable from the method copyist's 0.033 — and the two mechanisms then separate only by where they settle, 0.7002 against 0.8000, seven and a half spreads apart.
A grid on the wall is a scale without a projection
An Egyptian canon rules a wall into squares and counts a figure's height against the ruling — no horizon, no centre, and a length recovered to 1.4e-14% of error where the same reading taken off a pinhole misses by 58%. Applied instead to a pinhole picture, the furthest of six equal figures reads at 19% of its true height.
Higher on the page, and where that stops being true
A pinhole's image height above the horizon falls monotonically with depth over the whole 2–400 m range sampled here, and 57% of that whole range lands in the ground's last drawn tenth — the accumulation the oldest depth convention is quietly built from. Above eye level the ordering inverts, and at eye level exactly, five different depths draw one height, a spread of 0.0e+0 px.
No solid casts an aspective figure
Fitting the best single rigid view to an aspective figure — head and legs in profile, eye and shoulders turned square — misses its own marks by 2.6% of the drawn height, and no yaw does better than 3.0% in a full sweep. A genuine single-view drawing of the same body fits to 7.6e-13 pixels, and the five rotations recovered from the marks alone match the convention's own list to 0.0e+0°.
Two stations in one picture
A parallel floor under a perspective room found one map absorbing two centres into one sheet. Split a two-rule picture down the middle instead and each half hands back its own horizon — 7.80 px apart at a rule-mix of 0.02 — and no eye's position has anything to do with the gap, because an ordinary pinhole picture's recovered horizon does not depend on where the eye stood at all.
The room a divergent picture is a photograph of
A divergent construction depicts a rectangle on a plane leaning toward the camera, and the splay alone sets how far — 36.3° at a splay of 1.32. Stand a second such construction on the first one's far edge, as a wall, and the two recovered planes meet at 6.9°, not at the right angle a real room's corner would need.
What a removed wall costs that a removed roof does not
Fitting a single centre to a building with its near wall deleted lands at 3.0e-15 m — the arithmetic floor — because deleting a wall does not touch the projection, only which surfaces are drawn. Fitting the identical routine to the same building with its roof removed does not return a number at all: handed a bundle of genuinely parallel rays, it refuses outright.